Schrödinger equations with indefinite effective mass

C. Tezcan, R. Sever

Effective mass Schrödinger equation is solved exactly for a given potential. Nikiforov-Uvarov method is used to obtain energy eigenvalues and the corresponding wave functions. A free parameter is used in the transformation of the wave function. The effective mass Schrödinger equation is also solved for the Morse potential transforming to the constant mass Schrödinger equation for a potential. One can also get solution of the effective mass Schrödinger equation starting from the constant mass Schrödinger equation. PACS numbers: 03.65.-w; 03.65.Ge; 12.39.Fd Keywords: Position-dependent mass, Effective mass Schrödinger equation, Morse potential, Nikiforov-Uvarov method

K. J. Le Couteur

Abstract There are many problems of relativistic quantum mechanics which cannot be treated by working with a positive definite metric in the vector space of the fully quantized system. In this paper it is shown that, independently of the signs of energy and charge in the c number theory, any relativistic wave equation can be quantized and interpreted in terms of positive energies by use of an indefinite metric of suitable signature. A powerful method of obtaining positive definite transition probabilities from an indefinite metric is introduced and the concept of ‘negative probability’ is entirely avoided.The transition probabilities are defined as the squared matrix elements of a unitary matrix T, which is obtained from the solution of the Schrodinger equation by means of a special transformation.These techniques make it possible to treat relativistic wave equations which describe particles with several mass and spin states and to give a satisfactory account of the transition between two such mass states, a problem which is otherwise quite intractable. The relationship of spin and statistics is discussed and it is concluded that, apart from the special cases of spin 0, ½, 1, there is no necessary connexion between spin and statistics.

Olaf Stenzel

Daniel M. Greenberger

Using a formalism that includes particles with indefinite and variable masses, we can create a much richer set of states than in conventional quantum mechanics. For example, in a gravitational field the equivalence principle implies that the motion of a particle is independent of the mass. Thus it is possible to create physical states with large Δm but very small Δν and Δx, so that the trajectory of a particle is well determined, while the Δp · Δx uncertainty relation is satisfied. We also provide a quantitative description of wavepackets with arbitrary distributions in momentum and mass (or alternatively velocity and mass) and discuss a simple diffraction slit experiment as an example of their physical propagation. The Schrödinger equation is solved for a decaying particle at rest. It is also shown that Galilean invariance plays a natural and cohesive role within the variable mass formalism, unlike the case in conventional quantum mechanics where the existence of multiple mass states would violate Galilean invariance.